Skip to main content
SpringerLink
Log in

Existence of Kink Waves and Periodic Waves for a Perturbed Defocusing mKdV Equation

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

The existence of kink waves and periodic waves for a perturbed defocusing mKdV equation is established by using geometric singular perturbation theory. In addition, by analyzing the perturbation of the Hamiltonian vector field with an elliptic Hamiltonian of degree four, a two saddle cycle is exhibited. It is proven that the wave speed \(c_0(h)\) is decreasing on \(h\in [-3/4,0]\) by analyzing the ratio of Abelian integrals and the limit of wave speed is given. Furthermore, the relationship between the wave speed and the wavelength of traveling waves is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Korteweg, D.J., de Vries, G.: On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves. Philos. Mag. 39, 422 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive system. Philos. Trans. R. Soc. Lond. A 272, 47–78 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid. Mech. 78, 237–246 (1976)

    Article  MATH  Google Scholar 

  4. Camassa, R., Holm, D.: An integrable shallow wave equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Derks, G., Gils, S.: On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations. Jpn. J. Ind. Appl. Math. 10, 413–430 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ogama, T.: Travelling wave solutions to a perturbed Korteweg-de Vries equation. Hiroshima Math. J. 24, 401–422 (1994)

    MathSciNet  Google Scholar 

  7. Yan, W., Liu, Z., Liang, Y.: Existence of solitary waves and periodic waves to a perturbed generalized KdV equation. Math. Model. Anal. 19, 537–555 (2014)

    Article  MathSciNet  Google Scholar 

  8. Chen, A., Guo, L., Deng, X.: Existence of solitary waves and periodic waves for a perturbed generalized BBM equation. J. Differ. Equ. 261, 5324–5349 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Topper, J., Kawahara, T.: Approximate equations for long nonlinear waves on a viscous fluid. J. Phys. Soc. Jpn. 44, 663–666 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kuramoto, Y., Tsuzuki, T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55, 356–369 (1976)

    Article  Google Scholar 

  11. Sivashinsky, G.I.: Nonlinear analysis of hydrodynamic instability in laminar flames I. derivations of basic equations. Acta Astronaut. 4, 1177–1206 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, New York (1971)

    Book  MATH  Google Scholar 

  13. Angulo, J., Bona, J.L., Scialom, M.: Stability of cnoidal waves. Adv. Differ. Equ. 11, 1321–1374 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four: (I) saddle loop and two saddle cycle. J. Differ. Equ. 176, 114–157 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four: (II) cuspidal loop. J. Differ. Equ. 175, 209–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four: (III) global center. J. Differ. Equ. 188, 473–511 (2003)

    Article  MATH  Google Scholar 

  17. Dumortier, F., Li, C.: Perturbation from an elliptic Hamiltonian of degree four: (IV) figure eight-loop. J. Differ. Equ. 188, 512–554 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Carr, J., Chow, S.-N., Hale, J.K.: Abelian integrals and bifurcation theory. J. Differ. Equ. 59, 413–436 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chow, S.-N., Sanders, J.A.: On the number of critical points of the period. J. Differ. Equ. 64, 51–66 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cushman, R., Sanders, J.: A codimension two bifurcations with a third order Picard–Fuchs equation. J. Differ. Equ. 59, 243–256 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fan, X., Tian, L.: The existence of solitary waves of singularly perturbed mKdV-KS equation. Chaos Soliton Fract. 26, 1111–1118 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tang, Y., Xu, W., Shen, J., Gao, L.: Persistence of solitary wave solutions of singularly perturbed Gardner equation. Chaos Soliton Fract. 37, 532–538 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Choi, J.W., et al.: Multi-hump solutions of some singularly perturbed equations of KdV type. Discret. Contin. Dyn. A 34, 5181–5209 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cheng, C.Q., Küpper, T.: Dynamical behavior of two-soliton solution exhibited by perturbed sine-Gordon equation. Math. Nachr. 171, 53–77 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equation. J. Differ. Equ. 31, 53–98 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Carr, J.: Applications of the Center Manifold Theory. Applied Mathematieal Sciences, vol. 35. Springer, New York (1981)

    Book  Google Scholar 

  27. Chicone, C., Jacobs, M.: Bifurcation of critical periods for plane vector fields. Trans. Am. Math. Soc. 312, 433–486 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sabatini, M.: On the period function of Liénard systems. J. Differ. Equ. 152, 467–487 (1999)

    Article  MATH  Google Scholar 

  29. Zhao, Y.: The monotonicity of the period function for codimension four quadratic system \(Q_4\). J. Differ. Equ. 185, 370–387 (2002)

    Article  MATH  Google Scholar 

  30. Li, C., Lu, K.: The period function of hyperelliptic Hamiltonians of degree 5 with real critical points. Nonlinearity 21, 465–483 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gasull, A., Liu, C., Yang, J.: On the number of critical periods for planar polynomial systems of arbitrary degree. J. Differ. Equ. 249, 684–692 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen, X., Romanovski, V.G., Zhang, W.: Critical periods of perturbations of reversible rigidly isochronous centers. J. Differ. Equ. 251, 1505–1525 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Garijo, A., Villadelprat, J.: Algebraic and analytical tools for the study of the period function. J. Differ. Equ. 257, 2464–2484 (2014)

    Article  MATH  Google Scholar 

  34. Chen, A., Li, J., Huang, W.: The monotonicity and critical periods of periodic waves of the \(\phi ^6\) field model. Nonlinear Dyn. 63, 205–215 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Geyer, A., Villadelprat, J.: On the wave length of smooth periodic traveling waves of the Camassa–Holm equation. J. Differ. Equ. 259, 2317–2332 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work are supported by the National Natural Science Foundation of China (No. 11671107), Guangxi Natural Science Foundation (No. 2015GXNSFBA139004 and No. 2016GXNSFDA380031) and the Innovation Project of GUET Graduate Education (No.2016YJCX47).

Author information

Authors and Affiliations

  1. School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, 541004, People’s Republic of China

    Aiyong Chen & Lina Guo

  2. Department of Mathematics, Hunan First Normal University, Changsha, 410205, People’s Republic of China

    Aiyong Chen

  3. Department of Science, Guilin University of Aerospace Technology, Guilin, 541004, People’s Republic of China

    Wentao Huang

Authors
  1. Aiyong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lina Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wentao Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aiyong Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, A., Guo, L. & Huang, W. Existence of Kink Waves and Periodic Waves for a Perturbed Defocusing mKdV Equation. Qual. Theory Dyn. Syst. 17, 495–517 (2018). https://doi.org/10.1007/s12346-017-0249-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-017-0249-9

Keywords

Search

Navigation